Extended finite-element method for modeling the mechanical behavior of functionally graded material plates with multiple random inclusions

 Abstract— Functionally graded material is of great importance in many engineering problems. Here the effect of multiple random inclusions in functionally graded material (FGM) is investigated in this paper. Since the geometry of entire model becomes complicated when many inclusions with different sizes appearing in the body, a methodology to model those inclusions without meshing the internal boundaries is proposed. The numerical method couples the level set method to the extended finite-element method (X-FEM). In the X-FEM, the finite-element approximation is enriched by additional functions through the notion of partition of unity. The level set method is used for representing the location of random inclusions. Numerical examples are presented to demonstrate the accuracy and potential of this technique. The obtained results are compared with available refered results and COMSOL, the finite element method software.


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Abstract-Functionally graded material is of great importance in many engineering problems. Here the effect of multiple random inclusions in functionally graded material (FGM) is investigated in this paper. Since the geometry of entire model becomes complicated when many inclusions with different sizes appearing in the body, a methodology to model those inclusions without meshing the internal boundaries is proposed. The numerical method couples the level set method to the extended finite-element method (X-FEM). In the X-FEM, the finite-element approximation is enriched by additional functions through the notion of partition of unity. The level set method is used for representing the location of random inclusions. Numerical examples are presented to demonstrate the accuracy and potential of this technique. The obtained results are compared with available refered results and COMSOL, the finite element method software.
Index Terms-Extended finite-element method; multiple; random; inclusions, functionally graded material. FGM is a composite material whose mechanical properties change with a mathematical function. This function can contain a variable that is the coordinates of a point on an object. Because the material properties change throughout the body, FGM is of great interest in various technical fields. The main advantage of FGM is that there is no boundary between two different materials and therefore will not lead to discontinuous stress field in the body, despite the fact that the material properties may change drastically. In addition, the FGM can be created to optimize the stress distribution in the material. This is one of the new generation materials. In recent years, FGM has been used in most modern engineering disciplines, such as insulation in gas turbine engines, missile launchers, sensors, nanostructured materials and especially in space industry.
Functionally graded materials were used as alternative materials in some applications. Their special feature makes them very useful in reducing stress concentration but the exist of inclusion may reduce its stiffness. So, modeling the inclusion in FGM play a greatly important role in practice because they may cause the failure.
Most of the problem of discontinuous interfaces such as holes and inclusion is investigated with homogeneous material [1][2][3], [5] or the FGM structure contains only one defect such as void [4]. When modeling the interface problems by means of the finite element methods, the defect faces must be coincided with the edge of the elements and the FEM has encountered many Extended finite-element method for modeling the mechanical behavior of functionally graded material plates with multiple random inclusions Kim Bang Tran, The Huy Tran, Quoc Tinh Bui and Tich Thien Truong H difficulties. To overcome these difficulties, the extended finite element method (XFEM) was developed to solve those problems. In this paper, we present the XFEM for modeling multiple random inclusions in a finite FGM plate with arbitrarily varying elastic properties in the transverse direction. Poisson's ratio is held constant and Young's modulus is considered to vary across the radius and x-axis When multiple inclusions appear in objects with random sizes and positions, traditional finite element grids need to follow the profile of these particles. In the XFEM, the presence of inclusion does not alter the original element mesh. XFEM allows particle boundaries to cut through the mesh. The behavior of particles will be described by the enrichment function.

Level set method for inclusions detection
In XFEM, the level set method is used to detect discontinuous boundaries. According to [3], a boundary of an inclusion can be considered a material interface.
To calculate the normal level set function , consider Γ is the geometry of an inclusion. At any point x, we define the scattering point xΓ on the boundary so that the distance ||x -xΓ|| Is the smallest. The level set function  can be expressed as follows The appearance of inclusion with a particular boundary Γ can be detected by the level set value  as depicted in Fig. 1. In the whole body,  < 0 at any point located inside the domain bounded by  and  > 0 at any point located outside the domain.
With xc and rc is the center and radius of the impurity particle.

Enrichment functions for material discontinuities
To describe the physical properties of a material discontinuities element, we will use the absolute enrichment function as depicted in Fig. 2. According to [3], this function can be defined as the absolute value of the signed distance function as follow χ(x) can be calculated by interpolating the nodal signed distances within an element Where NI is shape function at node I Smoothing of χ away from the element layer containing the interface yields as shown in The derivative of absolute enrichment function with respect to x is given as shown in Fig. 4 x  I  I   I  I  I  I  I  I  I  I

XFEM for material interface
According to [3], the displacement field of a two-dimensional element with material discontinuity will be of the following form N is the total number of nodes and n is the number of nodes under the element; u is the transposing element at the nodes of the element as in the finite element method, a is the degree of freedom added at the enriched nodes and χ (x) is the enrichment function to describe the material discontinuity of the material boundary elements passing through.
With the stiffness matrix K of the enriched elements will be computed according to the formula As the FGM has the properties changing throughout the body, we need to divide the problem domain into a set of elements and obtained the information on node coordinates to take the Gaussian integral. Repeat on each Gauss point: compute the deformation matrix B at the Gauss point under consideration, compute the material matrix D at the point Gauss is considering, compute the element's stiffness matrix, assemble the element stiffness matrix into the global stiffness matrix.

Square plate with one circular inclusion
In the first example, a square plate with a circular inclusion is considered [1]. The model geometry and boundary conditions are described in Fig. 5. The plate side is L=5 m and the external load is q = 100 N/m. The lower edge of the plate is clamped. The matrix and inclusion materials are taken such as E1 = 3.10 7 N/m 2 , ν1=0.3 and E2 = 3.10 6 N/m 2 , ν2=0.25. Plane stress state is investigated. The XFEM mesh and enriched nodes are presented for this example in   We check the accuracy of the XFEM by comparing the obtained solutions with those given in previous work [1] as depicted in Fig. 7 and Fig.  8.
The ux displacement comparison is performed for the points along the horizontal red solid line of  We can see that the displacement results obtained by the XFEM matches well with those refered results [1], using GDQFEM.

FGM plate with material variation in the xdirection with seven circular inclusions
In the next example, we consider a rectangular isotropic FGM plate with material variation in the Cartesian x-direction, the dimensions and are depicted in Fig. 9. The external load is q = 1 N/m 2 . The lower edge of the plate is clamped and plane strain state is assumed. The plate contains seven circular inclusions. All inclusions have different radius and different positions as depicted in   We compare the finite element method (FEM) solution to that obtained by XFEM.  The computed results obtained by the XFEM and the FEM are listed in Table 2 including the percentage errors. The minimum and the maximum displacement obtained by the XFEM matches well with those derived from the FEM. The stress and displacement field of the plate are sketched in subsequent Fig. 10-12.

CONCLUSION
In this paper, an advanced of the XFEM is proposed for modeling multiple random inclusions in functionally graded material. It was observed that XFEM leads to very accurate results when compared with FEM and is suitable for solving discontinous problem when many inclusions with different sizes appear in the body.